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Canonical polyadic decomposition of thirdorder seminonnegative semisymmetric tensors using LU and QR matrix factorizations
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 150 (2014)
Abstract
Semisymmetric threeway arrays are essential tools in blind source separation (BSS) particularly in independent component analysis (ICA). These arrays can be built by resorting to higher order statistics of the data. The canonical polyadic (CP) decomposition of such semisymmetric threeway arrays allows us to identify the socalled mixing matrix, which contains the information about the intensities of some latent source signals present in the observation channels. In addition, in many applications, such as the magnetic resonance spectroscopy (MRS), the columns of the mixing matrix are viewed as relative concentrations of the spectra of the chemical components. Therefore, the two loading matrices of the threeway array, which are equal to the mixing matrix, are nonnegative. Most existing CP algorithms handle the symmetry and the nonnegativity separately. Up to now, very few of them consider both the seminonnegativity and the semisymmetry structure of the threeway array. Nevertheless, like all the methods based on line search, trust region strategies, and alternating optimization, they appear to be dependent on initialization, requiring in practice a multiinitialization procedure. In order to overcome this drawback, we propose two new methods, called ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$, to solve the problem of CP decomposition of seminonnegative semisymmetric threeway arrays. Firstly, we rewrite the constrained optimization problem as an unconstrained one. In fact, the nonnegativity constraint of the two symmetric modes is ensured by means of a square change of variable. Secondly, a Jacobilike optimization procedure is adopted because of its good convergence property. More precisely, the two new methods use LU and QR matrix factorizations, respectively, which consist in formulating highdimensional optimization problems into several sequential polynomial and rational subproblems. By using both LU and QR matrix factorizations, we aim at studying the influence of the used matrix factorization. Numerical experiments on simulated arrays emphasize the advantages of the proposed methods especially the one based on LU factorization, in the presence of highvariance model error and of degeneracies such as bottlenecks. A BSS application on MRS data confirms the validity and improvement of the proposed methods.
Introduction
Higher order (HO) arrays, commonly called tensors, play an important role in numerous applications, such as chemometrics [1], telecommunications [2], and biomedical signal processing [3]. They can be seen as HO extensions of vectors (oneway arrays) and matrices (twoway arrays). In many practical situations, the available data measurements cannot be arranged into a tensor form directly, that is to say, the observation diversity is insufficient either in time or frequency. However, if the latent data satisfies the statistical independence assumption, which is reasonable in many applications, meaningful HO arrays can be built by resorting to HO statistics (HOS) of the data [4]. In this instance, the HO arrays are partially symmetric or Hermitian due to the special algebraic structure of the basic HOS, such as moments and cumulants. In independent component analysis (ICA), the latent physical phenomena which are assumed to be statistically independent can be revealed by decomposing the HO array into factors. There exists several ways to decompose a given HO array, such as the Tucker model [5, 6]. Among the existing reliable HO array decomposition models, the canonical polyadic (CP) decomposition model has attracted much attention. Indeed, its uniqueness can be ensured under the sufficient conditions established by Kruskal [7]. In addition, unlike the HO singular value decomposition (HOSVD) [6], the CP model does not impose any orthogonality constraint on its factors.
Theoretically, a polyadic decomposition exactly fits an array by a sum of rankone terms [8]. A CP decomposition is defined as a polyadic decomposition with a minimal number of rankone terms which are needed to exactly fit a given HO array. Currently, the CP decomposition is gaining importance in several applications, for example, in exploratory data analysis [9], sensor array processing [10], telecommunications [11, 12], ICA [13], and in multipleinput multipleoutput radar systems [14]. A multitude of methods were developed to compute the CP decomposition. They include the iterative alternating least squares (ALS) procedure [15], which gains popularity due to its simplicity of implementation and low numerical complexity. Uschmajew proved the local convergence property of ALS under some conditions [16]. However, this convergence can be slow. Therefore, an enhanced line search (ELS) procedure was proposed by Rajih et al. [17] to cope with the slow convergence problem of ALS. Other approaches were also proposed, such as the conjugate gradient algorithm [18] and joint eigenvalue decompositionbased algorithms [19, 20], to cite a few. Some HO arrays enjoy certain properties, such as i) symmetry and ii) nonnegativity, which cannot be simply handled by the aforementioned general CP decomposition methods. Therefore, special CP models become more and more important.
The first special form of the CP model for threeway arrays that are symmetric in two modes brings the concept of individual differences in scaling (INDSCAL) analysis [21]. On one hand, INDSCAL analysis has been studied as a way of multiple factor analysis [22] with applications to chemometrics, psychology, and marketing research. On the other hand, in the domain of signal processing, and more particularly in blind source separation (BSS), the INDSCAL analysis is widely known as the joint diagonalization of a set of matrices by congruence (JDC). During the past two decades, many successful JDC methods have been proposed, such as Yeredor’s alternating columns and diagonal center (ACDC) algorithm [23], the joint approximate diagonalization (JAD) algorithm proposed by Cardoso and Souloumiac [24], the fast Frobenius diagonalization (FFDIAG) algorithm proposed by Ziehe et al. [25], Afsari’s LUJ1D algorithm [26], and many others [27–33]. A recent survey of JDC can be found in [34]. The second special form of CP model is defined when all the factors in the CP decomposition are constrained to be nonnegative, commonly known as nonnegative tensor factorization (NTF). NTF can be regarded as the extension of nonnegative matrix factorization (NMF) [35] to higher orders. In many applications, the physical properties are inherently nonnegative, such as chemistry [1] and fluorescence spectroscopy [36, 37]. In those applications, the results are only meaningful if the nonnegativity constraint is satisfied. Various methods for computing NTF and also NMF can be found in [38, 39].
So far, the CP model with both the symmetry and nonnegativity constraints has not received much attention. Coloigner et al. proposed a family of algorithms based on line search and trust region strategies [40]. Wang et al. developed an alternating minimization scheme [41]. Those methods appear to depend on initialization, and therefore in practice require a multiinitialization procedure, leading to an increase of numerical complexity. In this paper, we propose to fit the CP model of a threeway array by imposing both the seminonnegativity and the semisymmetry constraints. More precisely, we impose a nonnegativity constraint on the two symmetric modes of the INDSCAL model, which leads to the seminonnegative INDSCAL model or equivalently the CP decomposition of seminonnegative semisymmetric threeway arrays. Such a model is often encountered in ICA problems where a nonnegative mixing matrix is frequently considered. For example, in magnetic resonance spectroscopy (MRS), the columns of the mixing matrix represent the positive concentrations of the source metabolites. Then, the threeway array built by stacking the matrix slices of a cumulant array is both nonnegative and symmetric in two modes. In such a case, the seminonnegative INDSCAL problem is equivalent to the JDC problem subject to a nonnegativity constraint on the joint transformation matrix. We propose two new algorithms to solve the seminonnegative INDSCAL problem, called ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$. Firstly, we rewrite the constrained optimization problem as an unconstrained one. Actually, the nonnegativity constraint is ensured by means of a square change of variable. Secondly, we propose two Jacobilike approaches using LU and QR matrix factorizations, respectively, which consist in formulating highdimensional optimization problems into several sequential polynomial and rational subproblems. By using both LU and QR matrix factorizations, we aim at studying the influence of the used matrix factorization. Numerical experiments highlight the advantages of the proposed methods especially ${\text{JD}}_{\text{LU}}^{+}$, in the case of dealing with highvariance model error and with degeneracies such as bottlenecks. A BSS application on MRS signals confirms the validity and improvement of the proposed methods. A part of this work has been recently presented at the 8th IEEE Sensor Array and Multichannel Signal Processing Workshop [42].
The rest of the paper is organized as follows. After the presentation of some notations, the ‘Multilinear algebra prerequisites and problem statement’ section introduces some basic definitions of the multilinear algebra then gives the seminonnegative INDSCAL problem formulation. In the ‘Methods’ section, we describe the proposed algorithms in detail and also provide an analysis of the numerical complexities. The ‘Simulation results’ section shows the computer simulation results. Finally, we conclude the paper.
Multilinear algebra prerequisites and problem statement
Notations
The following notations are used throughout this paper. and denote the set of realvalued (N_{1} × N_{2} × ⋯ × N_{ i }) arrays and the set of nonnegative realvalued (N_{1} × N_{2} × ⋯ × N_{ i }) arrays, respectively. Vectors, matrices, and HO arrays are denoted by bold lowercase letters (a, b, ⋯), bold uppercase letters (A, B, ⋯) and bold calligraphic letters (, , ⋯), respectively. The (i,j)th entry of a matrix A is symbolized by A_{i,j}. Sometimes, the MATLAB^{®}; column/row notation is adopted to indicate submatrices of a given matrix or subarrays of a HO array. Also, a_{ i } denotes the ith column vector of matrix A. ⊡ denotes the Hadamard product (elementwise product), and ${\mathit{A}}^{\u22a12}=\mathit{A}\u22a1\mathit{A}$. ⊙ denotes the KhatriRao product. A^{♯} denotes the pseudo inverse of A. The superscripts ^{1}, ^{T}, and ^{T} stand for the inverse, the transpose, and the inverse after transpose operators, respectively. The (N × N) identity matrix is denoted by I_{ N }. 0_{ N } stands for Ndimensional vectors of zeros. a denotes the absolute value of a. ∥A∥_{ F } and $det\left(\mathit{A}\right)$ stand for the Frobenius norm and determinant of matrix A, respectively. diag(A) returns a matrix comprising only the diagonal elements of A. Diag(b) is the diagonal matrix whose diagonal elements are given by the vector b. off(A) vanishes the diagonal components of the input matrix A. vec(A) reshapes a matrix A into a column vector by stacking its columns vertically.
Definitions and problem formulation
Now we introduce some basic definitions in multilinear algebra which are necessary for the problem formulation.
Definition 1.
The outer product $\mathcal{C}={\mathit{u}}^{\left(1\right)}\circ {\mathit{u}}^{\left(2\right)}\circ {\mathit{u}}^{\left(3\right)}$ of three vectors is a threeway array of whose elements are defined by ${\mathcal{C}}_{{i}_{1},{i}_{2},{i}_{3}}={u}_{{i}_{1}}^{\left(1\right)}{u}_{{i}_{2}}^{\left(2\right)}{u}_{{i}_{3}}^{\left(3\right)}$.
Definition 2.
Each threeway array expressed as the outer product of three vectors is a rank1 threeway array.
More generally, the rank of a threeway array is defined as follows:
Definition 3.
The rank of an array , denoted by $\text{rk}\left(\mathcal{C}\right)$, is the minimal number of rank1 arrays belonging to that yield in a linear combination.
Despite the similarity between the definition of the tensor rank and its matrix counterpart, the rank of a threeway array may exceed its dimensions [4].
Definition 4.
A threeway array slice is a twodimensional section (fragment) of a threeway array, obtained by fixing one of the three indices [38].
For example, the kth frontal slice of a threeway array can be denoted by ${\mathcal{C}}_{:,:,k}$ using MATLAB notation, and sometimes it is also denoted by C^{(k)}.
The lowrank INDSCAL model of a threeway array is defined as follows:
Definition 5.
For a given P, corresponding to the number of rank1 terms, the INDSCAL model of a threeway array can be expressed as:
where the threeway array represents the model residual.
The notation $\mathcal{C}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right[\mathit{A},\mathit{A},\mathit{D}\left]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right]+\mathcal{V}$ refers to the INDSCAL decomposition (1) of with the associated loading matrices and $\mathit{D}=\left[{\mathit{d}}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{\mathit{d}}_{P}\right]\in {\mathbb{R}}^{K\times P}$. If and only if the residual is a null tensor, we have an exact INDSCAL decomposition.
An exact INDSCAL decomposition is considered to be essentially unique when it is only subject to scale and permutation indeterminacies. It means that an INDSCAL decomposition is insensitive to a scaling of the three vectors a_{ p }, a_{ p }, and d_{ p } provided that the product of the three scale numbers is equal to 1, and an arbitrary permutation of the rank1 terms. A necessary and sufficient uniqueness condition for the INDSCAL model was established by Afsari [43].
The INDSCAL model can also be described by using the frontal slices of :
where D^{(k)} is a diagonal matrix whose diagonal contains the elements of the kth row of D, and ${\mathit{V}}^{\left(k\right)}={\mathcal{V}}_{:,:,k}$.
In this paper, we propose to fit the INDSCAL model of threeway arrays while imposing nonnegativity constraints on both equal loading matrices A. It will be referred to as the seminonnegative INDSCAL model, as follows:
Problem 1.
Given and an integer P, find a seminonnegative INDSCAL model of $\mathcal{C}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right[\mathit{A},\mathit{A},\mathit{D}\left]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right]$, subject to the (N × P) matrix A having nonnegative components.
The seminonnegative INDSCAL problem is equivalent to the JDC problem subject to the nonnegativity constraint on the joint transformation matrix. In this paper, we mainly focus on the case of square nonnegative joint transformation matrix, for which N = P. The case of N > P will be discussed briefly in the next section. Therefore, the problem that we tackle in this paper is defined as follows:
Problem 2.
Given a threeway array with K symmetric frontal slices , find a (N × N) joint transformation matrix A and K diagonal matrices D ^{(k)} of dimension (N × N) such that:
by minimizing the residual term V^{(k)} in a leastsquares sense, subject to A having nonnegative components.
JDC cost functions
If the residual array is a realization of a Gaussian random array, it is logical to fit the INDSCAL model by the following direct least square (DLS) criterion [23, 44]:
and to minimize (4) with respect to A and D. Note that, in the field of ICA, only the loading matrix A is of interest since it corresponds to the mixing matrix of several latent source signals. The minimization of (4) with respect to D, when A is fixed, was given by Yeredor in [23]:
When A is orthogonal, we can replace D^{(k)} by $\text{Diag}\left\{{(\mathit{A}\odot \mathit{A})}^{\mathsf{\text{T}}}\phantom{\rule{2.77626pt}{0ex}}\text{vec}\left({\mathit{C}}^{\left(k\right)}\right)\right\}$ in (4). Then, the extra parameter D can be eliminated and the minimization of (4) is equivalent to minimizing the following indirect least square (IDLS) criterion [45, 46]:
In some cases such as in ICA, the orthogonality assumption of A can be satisfied by using a spatial whitening procedure [47]. However, it is known that the whitening procedure may introduce additional errors. Therefore, many algorithms propose to relax the orthogonality constraint by introducing the following cost function [25, 31]:
Frequently, the minimization of criterion (7) is performed on a matrix $\mathit{Z}\stackrel{\text{def}}{=}{\mathit{A}}^{\mathsf{\text{1}}}$ instead of A for simplicity, and Z is called the joint diagonalizer. To use this criterion, the matrix A (or Z) should be properly constrained in order to avoid the trivial zero solution and/or degenerate solutions [34].
Besides the criterions (4) and (7), Afsari [26] presented a new cost function, which is invariant to column scaling of A. Pham proposed an information theoretic criterion [48], which requires each matrix C^{(k)} to be positive definite. Tichavský and Yeredor gave a special weighted least square criterion [49].
Methods
Problem reformulation
Existing seminonnegative INDSCAL algorithms are based on the minimization of the cost function (4) [40, 41]. They are able to achieve a better estimation of A than ACDC when the data satisfies the seminonnegative INDSCAL model at the cost of a higher computational complexity. We propose to use criterion (7) based on elementary factorizations of A due to the fast convergence property of this kind of procedures. Generally, it is quite difficult to impose the nonnegativity constraint on A while computing its inverse A^{1} by minimizing (7). Let us consider the structure of $\mathcal{C}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right[\mathit{A},\mathit{A},\mathit{D}\left]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right]$ with the following assumptions:
① is nonsingular;
② does not contain zero entries.
Then, each frontal slice of is nonsingular and its inverse can be expressed as follows:
We use C^{(k,1)} to denote ${\left({\mathit{C}}^{\left(k\right)}\right)}^{1}$ for simplicity. Eq. 8 shows that C^{(k,1)} also preserves the jointly diagonalizable structure. Furthermore, instead of A^{1}, A serves as the joint diagonalizer. Then, A can be estimated by minimizing the following modified criterion based on (7):
By such a manipulation, most algorithms based on criterion (7) can now estimate A directly. However, none of them can guarantee the nonnegativity of A. In order to impose the nonnegativity constraint on A, we resort to use a square change of variable which was introduced by Chu et al. [50] for NMF, next adopted by Royer et al. for NTF [37] and by Coloigner et al. for seminonnegative INDSCAL [40]:
where . Then, problem 2 can be reformulated as follows:
Problem 3.
Given , find the square nonnegative loading matrix $\mathit{A}={\mathit{B}}^{\u22a12}$ such that B minimizes the following cost function:
LU and QR parameterizations of B
In order to minimize (11), one may consider a gradientlike approach. However, the performance of this kind of method is sensitive to the initial guess and to the search step size. In addition, the calculation of gradient of (11) with respect to B is computationally expensive due to the existence of the Hadamard product. Other algorithms, using Jacobilike procedures [25, 26, 31], parameterize A as a product of several special elementary matrices and estimate each elementary matrix successively. We propose to follow such a minimization scheme.
Now let us recall the following definitions and lemmas:
Definition 6.
A unit upper (or lower) triangular matrix is an upper (or lower, respectively) triangular matrix whose main diagonal elements are equal to 1.
Definition 7.
An elementary upper (or lower) triangular matrix with parameters {i,j,u_{i,j}} and i<j is a unit upper (or lower, respectively) triangular matrix whose nondiagonal elements are zeros except the (i,j)th entry, which is equal to u_{i,j}.
U^{(i,j)}(u_{i,j}) with 1 ≤ i < j ≤ N denotes an elementary upper triangular matrix:
Similarly, L^{(i,j)}(ℓ_{i,j}) with 1 ≤ j < i ≤ N corresponds to an elementary lower triangular matrix.
Definition 8.
A Givens rotation matrix with parameters {i,j,θ_{i,j}} and i < j is equal to an identity matrix except for the (i,i)th, (j,j)th, (i,j)th, and (j,i)th entries, which are equal to $\text{cos}\left({\theta}_{i,j}\right)$, $\text{cos}\left({\theta}_{i,j}\right)$, $\text{sin}\left({\theta}_{i,j}\right)$, and $\text{sin}\left({\theta}_{i,j}\right)$, respectively.
Q^{(i,j)}(θ_{i,j}) with 1 ≤ i < j ≤ N indicates the corresponding Givens rotation matrix:
Lemma 1.
Any (N × N) unit lower triangular matrix L whose (i,j)th component is ℓ_{i,j}(i > j) can be factorized as the following product of N (N1)/2 elementary lower triangular matrices [51, Chapter 3]:
where the two sets of indices ${J}_{1}$ and ${I}_{1}\left(j\right)$ are defined by ${J}_{1}=\{1,2,\dots ,N\}$ and ${I}_{1}\left(j\right)=\{j+1,j+2,\dots ,N\}$ for the sake of convenience. Similarly, any (N × N) unit upper triangular matrix U whose (i,j)th component is equal to u _{i,j} (i < j) can be factorized as a product of elementary upper triangular matrices as follows:
where ${I}_{2}$ and ${J}_{2}\left(i\right)$ are two sets of indices, defined by ${I}_{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\{N1,N2,\dots ,1\}$ and ${J}_{2}\left(i\right)\phantom{\rule{0.3em}{0ex}}=\{N,N1,\dots ,i+1\}$.
Lemma 2.
Any (N × N) orthonormal matrix Q can be factorized as the following product of N (N1)/2 Givens rotation matrices [52, Chapter 14]:
where ${I}_{2}$ and ${J}_{2}\left(i\right)$ are defined in Lemma 1.
For any nonsingular matrix , the LU matrix factorization decomposes it as B= L U Λ Π, where is a unit lower triangular matrix, is a unit upper triangular matrix, is a diagonal matrix, and is a permutation matrix. B also admits the QR matrix factorization as B= Q R Λ, where is an orthonormal matrix, is a unit upper triangular matrix, and is a diagonal matrix. Due to the indeterminacies of the JDC problem, the global minimum of (11), say B, can be expressed as B= L U and B= Q R without loss of generality. Moreover, by incorporating Lemma 1 and Lemma 2, we obtain the two following elementary factorizations of B:
As a consequence, the minimization of (11) with respect to B is converted to the estimate of N (N1) parameters: ℓ_{i,j} and u_{i,j} for the LU decomposition (17), or θ_{i,j} and u_{i,j} for the QR decomposition (18). Instead of simultaneously computing the N(N1) parameters, we propose two Jacobilike procedures which perform N(N1) sequential optimizations. This yields two new algorithms: i) the first algorithm based on (17), named ${\text{JD}}_{\text{LU}}^{+}$, estimates each ℓ_{i,j} and u_{i,j} successively, and ii) the second one based on (18), called ${\text{JD}}_{\text{QR}}^{+}$, estimates each θ_{i,j} and u_{i,j} sequentially.
Now, the difficulty is how to estimate four kinds of parameter, namely L^{(i,j)}(ℓ_{i,j}) and U^{(i,j)}(u_{i,j}) for ${\text{JD}}_{\text{LU}}^{+}$, and Q^{(i,j)}(θ_{i,j}) and U^{(i,j)}(u_{i,j}) for ${\text{JD}}_{\text{QR}}^{+}$. Two points should be noted here: i) L^{(i,j)}(ℓ_{i,j}) and U^{(i,j)}(u_{i,j}) belong to the same category of matrices; therefore, they can be estimated by the same algorithmic procedure just with an emphasis on the relation between the i and j indices (i < j for U^{(i,j)}(u_{i,j}) and j < i for L^{(i,j)}(ℓ_{i,j})); ii) for both ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms, the procedure of estimating U^{(i,j)}(u_{i,j}) is identical. Consequently, the principal problem is reduced to estimating two kinds of parameters, namely U^{(i,j)}(u_{i,j}) and Q^{(i,j)}(θ_{i,j}).
Minimization with respect to the elementary upper triangular matrix U^{(i,j)}(u_{i,j})
In this section, we minimize (11) with respect to U^{(i,j)}(u_{i,j}) with 1 ≤ i < j ≤ N. Let $\stackrel{~}{\mathit{A}}$ and $\stackrel{~}{\mathit{B}}$ denote the current estimate of A and B before estimating the parameter u_{i,j}, respectively. Let ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$ and ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$ stand for $\stackrel{~}{\mathit{A}}$ and $\stackrel{~}{\mathit{B}}$ updated by U^{(i,j)}(u_{i,j}), respectively. Furthermore, the update of $\stackrel{~}{\mathit{B}}$ is defined as follows:
In order to compute the parameter u_{i,j}, a typical way is to minimize the criterion (11) with respect to u_{i,j} by replacing matrix $\stackrel{~}{\mathit{B}}$ by ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$. For the sake of convenience, we denote J (u_{i,j}) instead of $J\phantom{\rule{0.3em}{0ex}}\left({\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}\right)$. Then, J(u_{i,j}) can be expressed as follows:
The expression of the Hadamard square of the update ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$ is shown in the following proposition:
Proposition 1.
${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}={\left({\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}\right)}^{\u22a12}={\left(\stackrel{~}{\mathit{B}}{\mathit{U}}^{(i,j)}\left({u}_{i,j}\right)\right)}^{\u22a12}$
can be expressed as a function of u _{i,j} as follows:
where ${\stackrel{~}{\mathit{b}}}_{i}$ and ${\stackrel{~}{\mathit{b}}}_{j}$ denote the ith and jth columns of $\stackrel{~}{\mathit{B}}$, respectively, and e_{ j } is the jth column of the identity matrix I_{ N }.
Inserting (21) into the cost function (20), we have:
where ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}={\stackrel{~}{\mathit{A}}}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}\stackrel{~}{\mathit{A}}$, ${\stackrel{~}{\mathit{c}}}^{(k,1)}=2\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{\mathit{A}}}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)$, ${\stackrel{~}{\mathit{c}}}^{(k,2)}=2\phantom{\rule{0.3em}{0ex}}{\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)}^{\mathsf{\text{T}}}\times {\mathit{C}}^{(k,1)}\stackrel{~}{\mathit{A}}$, and ${\stackrel{~}{c}}^{(k,3)}=4{\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)$ are a (N × N) constant matrix, a (N × 1) constant column vector, a (1 × N) constant row vector, and a constant scalar, respectively. The term ① in (22) transforms the jth column and the jth row of ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$. The term ② in (22) is a zero matrix except its jth column containing nonzero elements, while the term ③ contains nonzero entries only on its jth row. The term ④ is a zero matrix except its (j,j)th component being nonzero. In addition, ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}=\u2460+\u2461+\u2462+\u2463$ is a (N × N) symmetric matrix. Hence, (22) shows that only the jth column and jth row of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ involve the parameter u_{i,j}, while the other elements remain constant. Therefore, the minimization of the cost function (20) is equivalent to minimizing the sum of the squares of the jth columns of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ except their (j,j)th elements with k ∈ {1,⋯,K}. The required elements of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ can be expressed by the following proposition.
Proposition 2.
The elements of the jth column except the (j,j)th entry of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ is a seconddegree polynomial function in u _{i,j} as follows, for every value n different of j:
where ${\stackrel{~}{C}}_{n,i}^{\left(k\right)}$ and ${\stackrel{~}{C}}_{n,j}^{\left(k\right)}$ are the (n,i)th and (n,j)th components of matrix ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$, respectively, and ${\stackrel{~}{c}}_{n}^{(k,1)}$ is the nth element of vector ${\stackrel{~}{\mathit{c}}}^{(k,1)}$.
The proof of this proposition is straightforward. Indeed, we can show that the elements of the jth column except the (j,j)th entry of the term ① in (22) can be expressed by ${\stackrel{~}{C}}_{n,i}^{\left(k\right)}{u}_{i,j}^{2}+{\stackrel{~}{C}}_{n,j}^{\left(k\right)}$ with 1 ≤ n ≤ N and n ≠ j, and those elements of the term ② in (22) are equal to ${\stackrel{~}{c}}_{n}^{(k,1)}{u}_{i,j}$ with 1 ≤ n ≤ N and n ≠ j. The sum of these elements directly leads to (23). The terms ③ and ④ do not need to be considered, since they do not affect the offdiagonal elements in the jth column. Proposition 2 shows that the minimization of the cost function (20) can be expressed in the following compact matrix form:
where ${\mathit{Q}}_{E}=\sum _{k=1}^{K}{\left({\mathit{E}}^{\left(k\right)}\right)}^{\mathsf{\text{T}}}{\mathit{E}}^{\left(k\right)}$ is a (3 × 3) symmetric coefficient matrix. E^{(k)} is a ((N  1) × 3) matrix defined as follows: the first column contains the ith column of ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$ without the jth element, the second column contains vector ${\stackrel{~}{\mathit{c}}}^{(k,1)}$ without the jth entry, and the third column contains the jth column of ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$ without the jth component. ${\mathit{u}}_{i,j}={\left[{u}_{i,j}^{2},{u}_{i,j},1\right]}^{\mathsf{\text{T}}}$ is a threedimensional parameter vector.
Equation (24) shows that J(u_{i,j}) is a fourthdegree polynomial function. The global minimum u_{i,j} can be obtained by computing the roots of its derivative and selecting the one yielding the smallest value of (24). Once the optimal u_{i,j} is computed, ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$ is updated by (19) and the joint diagonalizer ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$ is updated by computing ${\left({\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}\right)}^{\u22a12}$. Then, the same procedure is repeated to compute the next u_{i,j} with another (i,j) index.
The minimization of (11) with respect to the elementary lower triangular matrix L^{(i,j)}(ℓ_{i,j}) with 1 ≤ j < i ≤ N can be computed in the same way. Proposition 2 is also valid for the parameter ℓ_{i,j} when 1 ≤ j < i ≤ N. The detailed derivation is omitted here. The processing of all the N (N  1) parameters u_{i,j} and ℓ_{i,j} is called a LU sweep. In addition, for estimating L^{(i,j)}(ℓ_{i,j}), the (i,j) index obeys the following order:
Regarding U^{(i,j)}(u_{i,j}), the (i,j) index varies according to the following sequence:
The proposed ${\text{JD}}_{\text{LU}}^{+}$ algorithm is comprised of several LU sweeps.
Minimization with respect to the Givens rotation matrix Q^{(i,j)}(θ_{i,j})
Now we minimize (11) with respect to Q^{(i,j)} (θ_{i,j}) with 1 ≤ i < j ≤ N. By abuse of notation, in this section, we continue to use $\stackrel{~}{\mathit{A}}$ and $\stackrel{~}{\mathit{B}}$ to denote the current estimate of A and B, respectively, before estimating the parameter θ_{i,j}. Also, let ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$ and ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$ stand for $\stackrel{~}{\mathit{A}}$ and $\stackrel{~}{\mathit{B}}$ updated by Q^{(i,j)}(θ_{i,j}), respectively. The update of $\stackrel{~}{\mathit{B}}$ is defined as follows:
Similarly, for computing the parameter θ_{i,j}, we can minimize the criterion (11) with respect to θ_{i,j} by replacing matrix $\stackrel{~}{\mathit{B}}$ by ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$. We denote J(θ_{i,j}) instead of $J\left({\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}\right)$ for convenience purpose. Then, J(θ_{i,j}) can be expressed as follows:
The Hadamard square of the update ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$ now can be rewritten as shown in the following proposition.
Proposition 3.
${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}={\left({\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}\right)}^{\u22a12}={\left(\stackrel{~}{\mathit{B}}{\mathit{Q}}^{(i,j)}\left({\theta}_{i,j}\right)\right)}^{\u22a12}$
can be written as a function of θ _{i,j} as follows:
where ${\stackrel{~}{\mathit{b}}}_{i}$ and ${\stackrel{~}{\mathit{b}}}_{j}$ denote the ith and jth columns of $\stackrel{~}{\mathit{B}}$, respectively, and e_{ i }and e_{ j }are the ith and jth columns of the identity matrix I_{ N }, respectively.
Inserting (29) into the cost function (28), we obtain:
where ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}={\stackrel{~}{\mathit{A}}}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}\stackrel{~}{\mathit{A}}$, ${\stackrel{~}{\mathit{c}}}^{(k,1)}={\stackrel{~}{\mathit{A}}}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)$, ${\stackrel{~}{\mathit{c}}}^{(k,2)}={\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}\stackrel{~}{\mathit{A}}$, and ${\stackrel{~}{c}}^{(k,3)}={\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}\left({\stackrel{~}{\mathit{b}}}_{i}\u22a1{\stackrel{~}{\mathit{b}}}_{j}\right)$ are a (N × N) constant matrix, a (N × 1) constant column vector, a (1 × N) constant row vector, and a constant scalar, respectively. The term ① in (30) transforms the ith and jth columns and the ith and jth rows of ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$. The term ② in (30) is a zero matrix except its ith and jth columns containing nonzero elements, while the term ③ contains nonzero entries only on its ith and jth rows. The term ④ is a zero matrix except its (i,i)th, (j,j)th, (i,j)th, and (j,i)th components being nonzero. ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}=\u2460+\u2461+\u2462+\u2463$ is a (N × N) symmetric matrix. Hence, (30) shows that only the ith and jth columns and the ith and jth rows of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ involve the parameter θ_{i,j}, while the other components remain constant. It is noteworthy that the (i,j)th and (j,i)th components are twice affected by the transformation. Considering the symmetry of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$, we propose to minimize the sum of the squares of the (i,j)th entries of the K matrices ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$, instead of minimizing all the offdiagonal entries. Although minimizing this quantity is not equivalent to minimizing the global cost function (28), such a simplified minimization scheme is commonly adopted in many algorithms, such as [20, 31]. We denote this local minimization by $\stackrel{~}{J}\left({\theta}_{i,j}\right)$. The (i,j)th component of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ is expressed in the following proposition.
Proposition 4.
The (i,j)th entry of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ can be expressed as a function of θ _{i,j} as follows:
where ${\stackrel{~}{C}}_{i,i}^{\left(k\right)}$, ${\stackrel{~}{C}}_{j,j}^{\left(k\right)}$, ${\stackrel{~}{C}}_{i,j}^{\left(k\right)}$, and ${\stackrel{~}{C}}_{j,i}^{\left(k\right)}$ are the (i,i)th, (j,j)th, (i,j)th, and (j,i)th components of matrix ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$, respectively. ${\stackrel{~}{c}}_{i}^{(k,q)}$ and ${\stackrel{~}{c}}_{j}^{(k,q)}$ are the ith and jth elements of vector ${\stackrel{~}{\mathit{c}}}^{(k,q)}$ with q ∈ {1,2}, respectively.
It is straightforward to show that the (i,j)th entry of the term ① in (30) can be expressed by $\stackrel{2}{\text{sin}}\left({\theta}_{i,j}\right)\stackrel{2}{\text{cos}}\left({\theta}_{i,j}\right)\left({\stackrel{~}{C}}_{i,i}^{\left(k\right)}+{\stackrel{~}{C}}_{j,j}^{\left(k\right)}\right)+\stackrel{4}{\text{sin}}\left({\theta}_{i,j}\right){C}_{j,i}^{\left(k\right)}+\stackrel{4}{\text{cos}}\left({\theta}_{i,j}\right){\stackrel{~}{C}}_{i,j}^{\left(k\right)}$, the (i,j)th element of the term ② is $\text{sin}\left(2{\theta}_{i,j}\right)\left(\phantom{\rule{0.3em}{0ex}}\stackrel{2}{\text{cos}}\left({\theta}_{i,j}\right){\stackrel{~}{c}}_{i}^{(k,1)}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\stackrel{2}{\text{sin}}\left({\theta}_{i,j}\right){\stackrel{~}{c}}_{j}^{(k,1)}\phantom{\rule{0.3em}{0ex}}\right)$, the (i,j)th component of the term ③ is equal to $\text{sin}\left(2{\theta}_{i,j}\right)\left(\phantom{\rule{0.3em}{0ex}}\stackrel{2}{\text{sin}}\left({\theta}_{i,j}\right)\right.{\stackrel{~}{c}}_{i}^{(k,2)}\phantom{\rule{0.3em}{0ex}}+\left(\right)close=")">\stackrel{2}{\text{cos}}\left({\theta}_{i,j}\right){\stackrel{~}{c}}_{j}^{(k,2)}$, and that of the term ④ is $\stackrel{2}{\text{sin}}\left(2{\theta}_{i,j}\right){\stackrel{~}{c}}^{(k,3)}$. Then, Proposition 4 can be proved.
In order to simplify the notation of (31), we resort to the Weierstrass change of variable: ${t}_{i,j}=\text{tan}\left({\theta}_{i,j}\right)$. Then, we obtain:
By substituting (32) into (31), we obtain an alternative expression of the (i,j)th entry of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ which is described in the following proposition. Then, the minimization of $\stackrel{~}{J}\left({\theta}_{i,j}\right)$ transforms to $\stackrel{~}{J}\left({t}_{i,j}\right)$.
Proposition 5.
The (i,j)th entry of ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$ can be expressed by a rational function of t _{i,j} as follows:
where ${f}_{4}^{\left(k\right)}={\stackrel{~}{C}}_{j,i}^{\left(k\right)}$, ${f}_{3}^{\left(k\right)}=2{\stackrel{~}{c}}_{i}^{(k,1)}$, ${f}_{2}^{\left(k\right)}={\stackrel{~}{C}}_{i,i}^{\left(k\right)}+{\stackrel{~}{C}}_{j,j}^{\left(k\right)}+2{\stackrel{~}{c}}_{j}^{(k,2)}4{\stackrel{~}{c}}^{(k,3)}$, ${f}_{1}^{\left(k\right)}=2{\stackrel{~}{c}}_{i}^{(k,2)}{\stackrel{~}{c}}_{j}^{(k,1)}$, and ${f}_{0}^{\left(k\right)}={\stackrel{~}{C}}_{j,j}^{\left(k\right)}$.
Eq. 33 easily shows that the sum of the squares of the (i,j)th entries of the K matrices ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$, is a rational function in t_{i,j}, namely $\stackrel{~}{J}\left({t}_{i,j}\right)$, where the degrees of the numerator and the denominator are 8 and 8, respectively. $\stackrel{~}{J}\left({t}_{i,j}\right)$ can be expressed in the following compact matrix form:
where ${\mathit{Q}}_{F}=\sum _{k=1}^{K}{\mathit{f}}^{\left(k\right)}{\left({\mathit{f}}^{\left(k\right)}\right)}^{\mathsf{\text{T}}}$ is a (5 × 5) symmetric coefficient matrix, ${\mathit{f}}^{\left(k\right)}={\left[{f}_{4}^{\left(k\right)},{f}_{3}^{\left(k\right)},{f}_{2}^{\left(k\right)},{f}_{1}^{\left(k\right)},{f}_{0}^{\left(k\right)}\right]}^{\mathsf{\text{T}}}$ is a fivedimensional vector, and τ_{i,j} is a fivedimensional parameter vector defined as follows:
The global minimum t_{i,j} can be obtained by computing the roots of its derivative and selecting the one yielding the smallest value of $\stackrel{~}{J}\left({t}_{i,j}\right)$. Once t_{i,j} is obtained, θ_{i,j} can be computed from the inverse tangent function ${\theta}_{i,j}=\text{arctan}\left({t}_{i,j}\right)$. It is noteworthy that the found θ_{i,j} cannot guarantee to decrease the actual cost function (28). If θ_{i,j} leads to an increase of (28), we reset θ_{i,j}= 0. Otherwise, ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$ is updated as described in (27) and the joint diagonalizer ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$ is updated by computing ${\left({\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}\right)}^{\u22a12}$. The same procedure will be repeated to compute θ_{i,j} with the next (i,j) index. The order of the (i,j) indices is defined in Eq. 26. The processing of all the N(N  1)/2 parameters θ_{i,j} and also the other N(N  1)/2 parameters u_{i,j} is called a QR sweep. Several QR sweeps yield the proposed ${\text{JD}}_{\text{QR}}^{+}$ algorithm.
Both of the ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms can be stopped when the value of cost function (11) or its relative change between two successive sweeps fall below a fixed small positive threshold. Such a stopping criterion is guaranteed to be met since the cost function is nonincreasing in each Jacobilike sweep.
Practical issues
In practice, we observe that if each frontal slice of the threeway array is almost exactly jointly diagonalizable due to a high signaltonoise ratio (SNR), the classical nonconstrained JDC methods can also give a nonnegative A with high probability. In this situation, the explicit nonnegativity constraint could be unnecessary and could increase the computational burden. Therefore, we propose to relax the nonnegativity constraint by directly decomposing A into elementary LU and QR forms, respectively, instead of using the decompositions of B as follows:
where the index sets ${I}_{1}\left(j\right)$, ${J}_{1}$, ${I}_{2}$, and ${J}_{2}\left(i\right)$ are defined in Lemma 1. By inserting (36) and (37) into the cost function (9), the ways of estimating the two sets of parameters {ℓ_{i,j},u_{i,j}} and {θ_{i,j},u_{i,j}} are identical to those of Afsari’s LUJ1D and QRJ1D methods [26], respectively. Therefore, in practice, in order to give an automatically SNRadaptive method, for ${\text{JD}}_{\text{LU}}^{+}$, in each Jacobilike iteration, we suggest to compute u_{i,j} by LUJ1D first. If all the elements in the jth column of $\stackrel{~}{\mathit{A}}{\mathit{U}}^{(i,j)}\left({u}_{i,j}\right)$ have the same sign ε, the update ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}=\epsilon \stackrel{~}{\mathit{A}}{\mathit{U}}^{(i,j)}\left({u}_{i,j}\right)$ is adopted. Otherwise, u_{i,j} is computed by minimizing (20) and ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$ is updated by computing (21). Each ℓ_{i,j} is computed similarly. Furthermore, the proposed ${\text{JD}}_{\text{QR}}^{+}$ and QRJ1D are combined in the same manner.
Afsari reported in [26] that if the rows of matrices ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$ (k ∈ {1,⋯,K}) are not balanced in their norms, the computation of the parameter could be inaccurate. In order to cope with this effect, we apply Afsari’s row balancing scheme every few sweeps. Such a scheme updates each ${\stackrel{~}{\mathit{C}}}^{(k,\text{new}))}$ by ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathit{\Lambda}{\stackrel{~}{\mathit{C}}}^{\left(k\right)}\phantom{\rule{0.3em}{0ex}}\mathit{\Lambda}$ and ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$ by ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\stackrel{~}{\mathit{A}}\mathit{\Lambda}$ using a diagonal matrix , whose diagonal elements are defined as follows:
where ${\stackrel{~}{\mathit{C}}}_{n,:}^{\left(k\right)}$ denotes the nth row of ${\stackrel{~}{\mathit{C}}}^{\left(k\right)}$.
In ICA, when a nonsquare matrix with N>P is encountered, the invertibility assumption of the frontal slices C^{(k)} does not hold. In this situation, we can compress A by means of a nonnegative matrix such that the resulting matrix $\stackrel{\u0304}{\mathit{A}}={\mathit{W}}_{\phantom{\rule{0.3em}{0ex}}+}\mathit{A}$ is a nonnegative square matrix. Then, the ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms can be used to compute the compressed loading matrix $\stackrel{\u0304}{\mathit{A}}$. W_{+} can be computed by using the nonnegative compression algorithm (NNCOMP) that we proposed in [53]. More precisely, given a realization of an observation vector, we obtain the square root of the covariance matrix, denoted by . The classical prewhitening matrix is computed by where ♯ denotes the pseudo inverse operator [47]. Then, the NNCOMP algorithm computes a linear transformation matrix such that W_{+} = Ψ W has nonnegative components. Once $\stackrel{\u0304}{\mathit{A}}$ is estimated, the original matrix A is obtained as follows:
It should be noted that generally A does not need to be computed in such an ICA problem, since the sources can be estimated directly by means of $\stackrel{\u0304}{\mathit{A}}$.
Numerical complexity
The numerical complexities of ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ are analyzed in terms of the number of floating point operations (flops). A flop is defined as a multiplication followed by an addition. In practice, only the number of multiplications, required to identify the loading matrix from a threeway array , is considered, which does not affect the order of magnitude of the numerical complexity.
For both algorithms, the inverses C^{(k,1)} (k ∈ {1,⋯,K}) of the frontal slices of cost N^{3}K flops, the initialization of ${\stackrel{~}{\mathit{C}}}_{\text{ini}}^{\left(k\right)}={\stackrel{~}{\mathit{A}}}_{\text{ini}}^{\mathsf{\text{T}}}{\mathit{C}}^{(k,1)}{\stackrel{~}{\mathit{A}}}_{\text{ini}}$ requires 2N^{3}K flops, and at each sweep, the calculation of parameters u_{i,j} needs N(N1)(5N^{2}+12N8)K/2 flops. In addition, in the case of the ${\text{JD}}_{\text{LU}}^{+}$ algorithm, the calculation cost of ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$, ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$, and ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$, with k ∈ {1,⋯,K}, is N (N1)(4N + (4 N + 1) K) flops, and the numerical complexity of computing the parameters ℓ_{i,j} is equal to that of u_{i,j}. Regarding the ${\text{JD}}_{\text{QR}}^{+}$ algorithm, for each sweep, the complexity of calculating the parameters θ_{i,j} is equal to N(N1)(5N^{2}+ 3 N + 29)K/2 flops, and the estimation of ${\stackrel{~}{\mathit{A}}}^{\left(\text{new}\right)}$, ${\stackrel{~}{\mathit{B}}}^{\left(\text{new}\right)}$, and ${\stackrel{~}{\mathit{C}}}^{(k,\text{new})}$, with k ∈ {1,⋯,K}, costs N (N  1)(5 N + (12 N + 20) K/2) flops. In practice, the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ techniques are combined with LUJ1D and QRJ1D [26], respectively, leading to the magnitude of global numerical complexities of ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ being between $\mathcal{O}\left({N}^{3}K\right)$ and $\mathcal{O}\left({N}^{4}K\right)$. A recent nonnegative JDC method called ${\text{ACDC}}_{\text{LU}}^{+}$[41] is also based on a square change of variable and LU matrix factorization. It minimizes the cost function (4) with respect to A and D alternately, leading to a higher numerical complexity. By means of the reformulation of the cost function, the proposed methods avoid the estimation of D, therefore achieving a lower complexity compared to ${\text{ACDC}}_{\text{LU}}^{+}$. The explicit expressions of the overall complexity of ${\text{JD}}_{\text{LU}}^{+}$, ${\text{JD}}_{\text{QR}}^{+}$, and ${\text{ACDC}}_{\text{LU}}^{+}$[41], as well as those of four classical JDC algorithms, namely ACDC [23], FFDIAG [25], LUJ1D [26], and QRJ1D [26], are listed in Table 1. One can notice that numerical complexities of the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ methods are at most one order of magnitude higher than those of the four JDC algorithms and still lower than that of ${\text{ACDC}}_{\text{LU}}^{+}$. Moreover, ${\text{JD}}_{\text{LU}}^{+}$ is less computationally expensive than ${\text{JD}}_{\text{QR}}^{+}$.
Simulation results
This section is twofold. In the first part, the performance of the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms is evaluated with simulated seminonnegative semisymmetric threeway arrays . Several experiments are designed to study the convergence property, the influence of SNR, the impact of the third dimension K of , the effect of the coherence of the loading matrix D, and the influence of the condition number of the diagonal matrices D^{(k)}. We also evaluate the proposed methods for estimating a nonsquare matrix A. The proposed algorithms are compared with four classical nonorthogonal JDC methods, namely ACDC [23], FFDIAG [25], LUJ1D [26], QRJ1D [26], and the nonnegative JDC method ${\text{ACDC}}_{\text{LU}}^{+}$[41]. In the second part, the source separation ability of the proposed algorithms is studied through a BSS application. In this context, the ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ are used to jointly diagonalize several matrix slices of the fourthorder cumulant array [40] of the observations and compared with several classical ICA [47, 54, 55] and NMF [56] methods.
Simulated seminonnegative INDSCAL model
The synthetic seminonnegative semisymmetric threeway array of rank N is generated randomly according to the seminonnegative INDSCAL model (3). When used without further specification, all the algorithms are manipulated under the following conditions:

i)
Model generation: The loading matrix is randomly drawn from a uniform distribution on the interval [ 0,1]. The loading matrix is drawn from a Gaussian distribution with a mean of 1 and a deviation of 0.5. The pure array is perturbed by a residual INDSCAL noise array . The loading matrices of are drawn from a zeromean unitvariance Gaussian distribution. The resulting noisy threeway array can be written as follows:
$${\mathcal{C}}_{N}=\frac{\mathcal{C}}{\parallel \mathcal{C}{\parallel}_{F}}+{\sigma}_{N}\frac{\mathcal{V}}{\parallel \mathcal{V}{\parallel}_{F}}$$(40)where σ_{ N } is a scalar controlling the noise level. Then, the SNR is defined by $\text{SNR}=20\underset{10}{\text{log}}\left({\sigma}_{N}\right)$.

ii)
Initialization: In each Monte Carlo trial, all the algorithms are initialized by a same random matrix whose components obey the uniform distribution over [ 0,1].

iii)
Afsari’s row balancing scheme: The LUJ1D, QRJ1D, ${\text{JD}}_{\text{LU}}^{+}$, and ${\text{JD}}_{\text{QR}}^{+}$ algorithms perform the row balancing scheme once per run of five sweeps.

iv)
Stopping criterion: All the algorithms stop either when the relative error of the corresponding criterion between two successive sweeps is less than 10^{5} or when the number of sweeps exceeds 200. A sweep of ACDC includes a full AC phase and a DCphase.

v)
Performance measurement: The performance is measured by means of the error between the true loading matrix A and the estimate $\stackrel{~}{\mathit{A}}$, the numerical complexity, and the CPU time. We define the following scaleinvariant and permutationinvariant distance [40]:
$$\alpha (\mathit{A},\stackrel{~}{\mathit{A}})=\frac{1}{N}\sum _{n=1}^{N}\phantom{\rule{1em}{0ex}}\underset{(n,{n}^{\prime})\in {I}_{n}^{2}}{min}d({\mathit{a}}_{n},{\stackrel{~}{\mathit{a}}}_{{n}^{\prime}})$$(41)where a_{ n } and ${\stackrel{~}{\mathit{a}}}_{{n}^{\prime}}$ are the nth column of A and the n^{′}th column of $\stackrel{~}{\mathit{A}}$, respectively. ${I}_{n}^{2}$ is defined recursively by ${I}_{1}^{2}=\{1,\cdots \phantom{\rule{0.3em}{0ex}},N\}\times \{1,\cdots \phantom{\rule{0.3em}{0ex}},N\}$, and ${I}_{n+1}^{2}={I}_{n}^{2}{J}_{n}^{2}$ where ${J}_{n}^{2}={\text{argmin}}_{(n,{n}^{\prime})\in {I}_{n}^{2}}d({\mathit{a}}_{n},{\stackrel{~}{\mathit{a}}}_{{n}^{\prime}})$. In addition, $d({\mathit{a}}_{n},{\stackrel{~}{\mathit{a}}}_{{n}^{\prime}})$ is defined as the pseudodistance between two vectors [13]:
$$d({\mathit{a}}_{n},{\stackrel{~}{\mathit{a}}}_{{n}^{\prime}})=1\frac{{\u2225{\mathit{a}}_{n}^{\mathsf{\text{T}}}\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{\mathit{a}}}_{{n}^{\prime}}\u2225}^{2}}{\parallel {\mathit{a}}_{n}{\parallel}^{2}\parallel {\stackrel{~}{\mathit{a}}}_{{n}^{\prime}}{\parallel}^{2}}$$(42)The criterion (41) is an upper bound of the optimal permutationinvariant criterion. It avoids the burdensome computation of all the permutations. A small value of (41) means a good performance in the sense that $\stackrel{~}{\mathit{A}}$ is close to A.

vi)
Test environment: The simulations are carried out in Matlab v7.14 on Mac OS X and run on Intel QuadCore CPU 2.8 GHz with 32 GB memory. Moreover, we repeat all the experiments with 500 Monte Carlo trials.
Convergence
In this experiment, the convergences of the ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms are compared to those of ACDC, FFDIAG, LUJ1D, QRJ1D, and ${\text{ACDC}}_{\text{LU}}^{+}$. The dimensions of the threeway array are set to N = 5 and K = 15. The performance is assessed under three SNR conditions: SNR =  5, 10, and 25 dB, respectively. Figure 1 shows the convergence curves measured in terms of the cost function as a function of sweeps. It shows that FFDIAG, LUJ1D, and QRJ1D exhibit fast convergence behavior. They converge in less than 20 sweeps. ${\text{ACDC}}_{\text{LU}}^{+}$ decreases the cost function (4) quasilinearly. ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$ do not converge in a maximum of 200 sweeps. The proposed ${\text{JD}}_{\text{LU}}^{+}$ algorithm converges in about 100 sweeps when SNR =25 dB and SNR =10 dB, and it converges in about 40 sweeps when SNR =5 dB. Regarding ${\text{JD}}_{\text{QR}}^{+}$, it reduces the cost function (11) to the values relatively higher than those achieved by ${\text{JD}}_{\text{LU}}^{+}$ and converges in about 50 sweeps whatever the SNR is. It seems that FFDIAG, LUJ1D, and QRJ1D achieve the fastest convergence rate. It should be noted that while an algorithm may converge to a point in which the value of the cost function is close to zero, such a point could be a local minimum far from the desired matrix A as shown in Figure 2. The top picture in Figure 2 shows the convergence curves measured in terms of the estimating error $\alpha (\mathit{A},\stackrel{~}{\mathit{A}})$ as a function of sweeps when SNR=25 dB. It shows that the solutions of FFDIAG, LUJ1D, and QRJ1D are still far from optimum. ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$ give better estimations of A than the previous three methods. The best results are achieved by the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ methods. The middle picture in Figure 2 displays the convergence curves when SNR=10 dB. It can be observed that ACDC converges to a local minimum which is not the global one and that the performance of the proposed methods is still better than that of the five other algorithms. For a low SNR =5 dB, as shown in the bottom picture in Figure 2, both the methods based on alternating optimization, namely ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$, converge to local minima which are less desirable. The proposed algorithms are always able to converge to better results than the classical methods. The average numerical complexities and CPU time of all the algorithms over Monte Carlo trials are shown in Table 2. It is observed that FFDIAG, LUJ1D, and QRJ1D require a small amount of calculations, whereas ${\text{ACDC}}_{\text{LU}}^{+}$ requires a large amount of calculations. The proposed ${\text{JD}}_{\text{LU}}^{+}$ just costs a bit more flops and CPU time than ACDC, but it is still much more efficient. Concerning the ${\text{JD}}_{\text{QR}}^{+}$ algorithm, it is more costly than ${\text{JD}}_{\text{LU}}^{+}$, with a comparable performance. We can then conclude that ${\text{JD}}_{\text{LU}}^{+}$ offers the best performance/complexity compromise in these experiments.
Effect of SNR
In this section, we study the behaviors of the seven algorithms as a function of SNR. The dimensions of the threeway array ${\mathcal{C}}_{N}$ are set to N = 5 and K = 15. We repeat the experiments with SNR ranging from 30 to 50 dB with a step of 2 dB. The top picture in Figure 3 depicts the average curves of $\alpha (\mathit{A},\stackrel{~}{\mathit{A}})$ of the seven algorithms as a function of SNR. The obtained results show that the performance of all the methods increases as SNR grows. For the unconstrained methods, generally, ACDC performs better than FFDIAG, LUJ1D, and QRJ1D. The nonnegativity constraint obviously helps ${\text{ACDC}}_{\text{LU}}^{+}$, ${\text{JD}}_{\text{LU}}^{+}$, and ${\text{JD}}_{\text{QR}}^{+}$ to improve the results for lower SNR values. The performance of ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$ remains stable for higher SNR values due to the small number of available sweeps and the lack of good initializations. Generally, the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms outperform the others when SNR is between 20 and 30 dB and perform similar to FFDIAG, LUJ1D, and QRJ1D when SNR is above 45 dB. The average numerical complexity and CPU time at each SNR level of all the methods in this experiment are shown in the bottom of Figure 3. It shows that the proposed methods achieve better estimations of A and cost less flops and CPU time than ${\text{ACDC}}_{\text{LU}}^{+}$. The ${\text{JD}}_{\text{LU}}^{+}$ gives the best performance/complexity tradeoff for all the considered SNR values.
Effect of dimension K
In ICA, the third dimension K of the threeway array corresponds to the number of covariance matrices at different lags, or the number of matrix slices derived from a cumulant array. In this section, we study the influence of K on the performance of the seven algorithms. The first and second dimensions of ${\mathcal{C}}_{N}$ are set to N = 5. The SNR value is fixed to 10 dB. We repeat the experiment with K ranging from 3 to 55. The top picture in Figure 4 shows the average curves of $\alpha (\mathit{A},\stackrel{~}{\mathit{A}})$ of all the algorithms as a function of K. For the five existing methods, ACDC, ${\text{ACDC}}_{\text{LU}}^{+}$, FFDIAG, LUJ1D, and QRJ1D, their performance is quite stable with respect to K. The performance of the proposed methods progresses as K increases and then practically stabilizes for high values of K. It indicates that after some point (e.g., K ≥ 20), the additional information brought by an increase of K does not further improve the results. The proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms maintain competitive advantages through all the K values. The two images in the bottom of Figure 4 present the average numerical complexity and CPU time of all the algorithms in this experiment, respectively. It shows that the numerical complexity of ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ is between that of ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$. The ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ methods seem to be the most effective algorithms compared to the other methods.
Effect of coherence of D
In this experiment, the effect of the coherence of the third loading matrix D of the threeway array $\mathcal{C}=\left[\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right[\mathit{A},\mathit{A},\mathit{D}\left]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right]$ is evaluated. Let d_{ n } and d_{ m } denote the nth and mth columns of D, respectively. The angle ψ_{n,m} between d_{ n } and d_{ m } can be derived by using the following Euclidean dot product formula ${\mathit{d}}_{n}^{\mathsf{\text{T}}}\phantom{\rule{0.3em}{0ex}}{\mathit{d}}_{m}=\parallel {\mathit{d}}_{n}\parallel \parallel {\mathit{d}}_{m}\parallel \text{cos}\left({\psi}_{n,m}\right)$. Then, the coherence ρ of D is defined as the maximum absolute cosine of angle ψ_{n,m} between the columns of D as follows:
The quantity ρ is also known as the modulus of uniqueness of JDC [43]. By its definition (43), ρ falls in the range of [ 0,1]. The JDC problem is considered to be illconditioned when ρ is close to 1. Such an illconditioned problem can be met in ICA when A has nearly collinear column vectors. For example, in order to perform ICA, provided that all the sources are nonGaussian, which is often the case in practice, we can build a threeway array by stacking the matrix slices of the fourthorder cumulant array of the observation data. Then, the loading matrix D can be expressed as follows:
where ${\mathit{C}}_{4,\left\{\mathit{s}\right\}}=\text{diag}\phantom{\rule{0.3em}{0ex}}\left[{\mathcal{C}}_{1,1,1,1,\left\{\mathit{s}\right\}},\cdots \phantom{\rule{0.3em}{0ex}},{\mathcal{C}}_{N,N,N,N,\left\{\mathit{s}\right\}}\right]$ is a (N × N) diagonal matrix with ${\mathcal{C}}_{n,n,n,n,\left\{\mathit{s}\right\}}$ being the fourthorder cumulant of the nth source, n ∈ {1,⋯,N}, and where ⊙ denotes the KhatriRao product. It can be observed that the coherence of the columns of A will influence the coherence of the matrix D. In the following test, the dimensions of the threeway array ${\mathcal{C}}_{N}$ are set to N = 5 and K = 15. The SNR value is fixed to 10 dB. In order to control ρ, firstly, we randomly generate an orthogonal matrix so that ρ = 0 by orthogonalizing a (15×5) random matrix. Secondly, we rotate its five columns such that all the internal angles between any columns are equal to a predefined value ψ. Therefore, ρ is only controlled by the angle ψ and equals to $\left\text{cos}\right(\psi \left)\right$. We repeat the experiment with the angle ψ ranging from 0 to π/2 with a step of π/60. A small ψ value means a large ρ value. The top picture in Figure 5 displays the average curves of $\alpha (\mathit{A},\stackrel{~}{\mathit{A}})$ of all the algorithms as a function of ψ. It shows that the nonnegativity constrained methods ${\text{ACDC}}_{\text{LU}}^{+}$, ${\text{JD}}_{\text{LU}}^{+}$, and ${\text{JD}}_{\text{QR}}^{+}$, outperform the unconstrained ones ACDC, FFDIAG, LUJ1D, and QRJ1D. The proposed algorithms are more efficient, particularly when the coherence level is high. The average numerical complexity and CPU time displayed in the bottom of Figure 5 indicate that the ${\text{JD}}_{\text{LU}}^{+}$ algorithm provides the best performance/complexity compromise, while the ${\text{JD}}_{\text{QR}}^{+}$ algorithm is also competitive with regard to ${\text{ACDC}}_{\text{LU}}^{+}$.
Effect of condition number of D ^{(k)}
When the JDC problem is considered, a diagonal matrix D^{(k)} could contain some diagonal elements which, despite being nonzero, are many orders of magnitude lower than some other elements, leading to an illconditioned matrix C^{(k)}. For the proposed methods, the inverse of such a matrix C^{(k)} would contain numerical errors. In this experiment, we study the performance of the seven algorithms as a function of the condition number of one of the diagonal matrices D^{(k)}. The dimensions of the threeway array ${\mathcal{C}}_{N}$ are set to N = 5 and K = 15. The SNR value is set to 10 dB. We vary the condition number of the first diagonal matrix D^{(1)} from 1 to 1,000 by fixing the ratio of its largest diagonal element to its smallest diagonal element. The top picture in Figure 6 displays the average curves of the estimating error $\alpha (\mathit{A},\stackrel{~}{\mathit{A}})$ of the seven algorithms as a function of the condition number of D^{(1)}. The results reveal that a highly illconditioned diagonal matrix D^{(1)} has a clear negative effect on the estimation accuracy of all the algorithms. The nonnegativity constrained methods ${\text{ACDC}}_{\text{LU}}^{+}$, ${\text{JD}}_{\text{LU}}^{+}$, and ${\text{JD}}_{\text{QR}}^{+}$ outperform the classical algorithms ACDC, FFDIAG, LUJ1D, and QRJ1D whatever the condition number is. The proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms maintain advantages when the condition number is less than 100. Regarding the cases of larger condition numbers, ${\text{ACDC}}_{\text{LU}}^{+}$ is more superior since it does not need to invert the highly illconditioned matrix. It is worthy pointing out that in practice, we can choose these sufficiently wellconditioned matrices C^{(k)} for the proposed methods, whose condition numbers are below a predefined threshold. In addition, a weighted cost function for which weights would depend on the condition number of each matrix can be considered. On the other hand, the performance of the classical methods can also be improved by choosing a particular subset of available matrices [57] and by properly weighting the cost functions [49]. In order to give a fair comparison, all the algorithms operate on the same set of matrices in all the experiments of this paper. In addition, the average numerical complexity and CPU time at each condition number of all the methods in this experiment are shown in the bottom of Figure 6. It shows that the proposed methods give the best performance/complexity tradeoff compared to ${\text{ACDC}}_{\text{LU}}^{+}$ whatever the condition number is.
Test with a nonsquare matrix A
As described in the section of practical issues, when a nonsquare matrix with N>P is met in ICA, we propose to compress it by a nonnegative compression matrix [53], such that the resulting matrix $\stackrel{\u0304}{\mathit{A}}={\mathit{W}}_{\phantom{\rule{0.3em}{0ex}}+}\mathit{A}$ is a (P × P) nonnegative square matrix. Then, the proposed methods can be applied to estimate $\stackrel{\u0304}{\mathit{A}}$. Similar to the classical prewhitening, the nonnegative compression step could introduce numerical errors. In this experiment, we compare our methods to ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$ through a simulated ICA model. The latter algorithms can directly estimate a nonsquare matrix A from the fourthorder cumulant matrix slices. The ICA model is established as follows:
where $\mathit{x}\left[f\right]={\left[{x}_{1}\left[f\right],\cdots \phantom{\rule{0.3em}{0ex}},{x}_{N}\left[f\right]\right]}^{\mathsf{\text{T}}}$ is the (N × 1) observation vector, $\mathit{s}\left[f\right]={\left[{s}_{1}\left[f\right],{s}_{2}\left[f\right],{s}_{3}\left[f\right]\right]}^{\mathsf{\text{T}}}$ is the (3 × 1) zeromean unitvariance source vector whose elements are independently drawn from a uniform distribution over $\left[\sqrt{3},\sqrt{3}\right]$, $\mathit{\nu}={\left[{\nu}_{1}\left[f\right],\cdots \phantom{\rule{0.3em}{0ex}},{\nu}_{N}\left[f\right]\right]}^{\mathsf{\text{T}}}$ is the (N × 1) zeromean unitvariance Gaussian noise vector, and A is the (N × 3) nonnegative mixing matrix whose components are independently drawn from a uniform distribution over [ 0,1]. In this context, the SNR is defined by:
For the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms, the given realization of {x[f] } is compressed by means of a matrix computed using the method proposed in [53], leading to a threedimensional vector $\left\{\stackrel{\u0304}{\mathit{x}}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}$. We compute the fourthorder cumulant array of $\left\{\stackrel{\u0304}{\mathit{x}}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}$ and choose the first three matrix slices in order to build a threeway array. Hence, ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ decompose a (3×3×3) array. Once the compressed mixing matrix $\stackrel{\u0304}{\mathit{A}}$ is estimated, the original mixing matrix is obtained by Eq. 39. Regarding ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$, the fourthorder cumulant array of {x[f] } is directly computed without compression. We apply ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$ on two threeway arrays with different third dimensions. The first array of dimension (N × N × 3) is built by choosing the first three matrix slices from the fourthorder cumulant array, while the second array of dimension (N × N× N) is built using the first N matrix slices. We study the impact of the number of observations N on the performance of the JDC algorithms, by varying N from 4 to 24. The SNR value is fixed to 5 dB. The number of samples used to estimate the cumulants is set to 10^{3}. Figure 7 shows the average curves of the estimating error $\alpha (\mathit{A},\stackrel{~}{\mathit{A}})$ of all the algorithms as a function of N. As it can be seen, when N ≤ 15, the larger the value of N, the more accurate estimation of A is achieved. When N > 15, the further increase of N does not bring significant improvement in terms of the estimation accuracy. ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$ give better results when the array with a larger third dimension is considered. Their results on (N × N × N) arrays outperform the proposed methods when N = 4. ${\text{ACDC}}_{\text{LU}}^{+}$ also gives the best estimation on (N × N × N) arrays with N = 5. It suggests that the numerical errors introduced by the compression step limit the performance of the proposed methods when only a small number of observation is available. Such a negative effect can be partially compensated by using a large number of observations, since the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ methods maintain the highest performance in terms of estimation accuracy when N≥6. The performance ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$ can be further improved by using a (N × N ×N^{2}) array, which contains all the N^{2} fourthorder cumulant matrix slices. However, it leads to a higher numerical complexity especially for a large value of N. Regarding the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ methods, their performance can also be improved by using all the nine matrix slices of the fourthorder cumulant array of the compressed observation vector. Nevertheless, the experimental result has already shown that by using only a small number of matrix slices, ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ can maintain lower numerical complexities than ACDC and ${\text{ACDC}}_{\text{LU}}^{+}$, while achieving better estimation results, when a large value of N is considered. Therefore, despite the negative influence of the nonnegative compression, the proposed methods still offer a good performance/complexity compromise to estimate a nonsquare matrix A.
BSS application on MRS data
In this section, we aim to illustrate the potential capability of the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms for solving a reallife BSS problem by an application carried on simulated MRS data.
MRS is a powerful noninvasive analytical technique for analyzing the chemical content of MRvisible nuclei and therefore enjoys particular advantages for assessing metabolism. The chemical property of each nucleus determines the frequency at which it appears in the MR spectrum, giving rise to peaks corresponding to specific metabolites [58]. Therefore, the MRS observation spectra can be modeled as the mixture of the spectrum of each constitutional source metabolite. More specifically, it can be written as the noisy linear instantaneous mixing model described in Eq. 45, where is the MRS observation vector, is the source vector representing the statistically quasiindependent source metabolites, is the instrumental noise vector, and is the nonnegative mixing matrix containing the positive concentrations of the source metabolites. SNR is defined as in Eq. 46. In this experiment, two simulated MRS source metabolites {s_{1}[f] } and {s_{2}[f] }, namely the Choline (Cho) and Myoinositol (Ins) (see Figure 8b), are generated by Lorentzian and Gaussian functions [59]. Each of the sources contains 10^{3} samples. The observation vector x[f] is generated according to (45). The components of the (N×2) mixing matrix A are randomly drawn from a uniform distribution. The additive noise ν[f] is modeled as a zeromean unitvariance Gaussian vector. The ICA methods based on the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms, namely ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA, consist of four steps: i) compressing {x[f] } by means of a matrix [53], ii) estimating the fourthorder cumulant array of the compressed observations and stacking all the cumulant matrix slices in a threeway array, iii) decomposing the resulting threeway array by means of ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$, respectively, and iv) reconstructing the sources. The ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA are compared to four stateoftheart BSS algorithms, namely two efficient ICA methods CoM _{2}[54] and SOBI [47], the nonnegative ICA (NNICA) method with a line search along the geodesic [55], and the NMF method [56] based on alternating nonnegativity least squares. The performance is assessed by means of the error $\alpha \phantom{\rule{0.3em}{0ex}}\left({\left\{\mathit{s}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}}^{\mathsf{\text{T}}},{\left\{\stackrel{~}{\mathit{s}}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}}^{\mathsf{\text{T}}}\right)$ between the true source s[f] and its estimate $\stackrel{~}{\mathit{s}}\left[f\right]$, the numerical complexity, and the CPU time. To find out the detailed analysis of the numerical complexity of the classical ICA algorithms, the reader can refer to the book chapter [60]. Figure 8 shows an example of the separation results of all the methods with N = 32 observations and a SNR of 10 dB. Regarding CoM _{2}, SOBI, NNICA, and NMF, there are some obvious disturbances presented in the estimated metabolites. As far as ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA are concerned, the estimated source metabolites are quasiperfect. Furthermore, the comprehensive performance of all the methods will be studied by the following experiments with 200 independent Monte Carlo trials.
In the first experiment, the effect of the number of observations N is evaluated. The SNR is fixed to 10 dB. The six methods are compared with N ranging from 4 to 116 with a step of 4. The average curves of error $\alpha \phantom{\rule{0.3em}{0ex}}\left({\left\{\mathit{s}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}}^{\mathsf{\text{T}}},{\left\{\stackrel{~}{\mathit{s}}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}}^{\mathsf{\text{T}}}\right)$ as a function of N are shown in the left image of Figure 9. It can be seen that the estimating errors of all the methods improve as N increases. It suggests that in noisy BSS contexts, using more sensors often yields better results. The proposed ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA methods maintain the competitive advantages. The average curves of the numerical complexities of this experiment are shown in the bottom left picture of Figure 9. We can notice that the numerical complexities of all the methods increase with N. The complexities of ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA seem identical in the logarithmic scaled plot, which is because theoretically their complexities are mainly dominated by the computation of the nonnegative compression step and of the cumulants. Indeed, ${\text{JD}}_{\text{LU}}^{+}$ICA is more computationally efficient than ${\text{JD}}_{\text{QR}}^{+}$ICA in the step of CP decomposition of the cumulant array. This can be verified by the average CPU time of those methods, shown in the bottom right image of Figure 9. We can observe that ${\text{JD}}_{\text{LU}}^{+}$ICA is slower than CoM _{2}, but it is faster than NNICA, SOBI, and NMF.
In the second experiment, we study the influence of SNR on the performance of the six methods. The number of observations N is set to 32. SNR is varied from 0 to 50 dB with a step of 2 dB. The average curves of the estimating error $\alpha \phantom{\rule{0.3em}{0ex}}\left({\left\{\mathit{s}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}}^{\mathsf{\text{T}}},{\left\{\stackrel{~}{\mathit{s}}\right[f\left]\phantom{\rule{0.3em}{0ex}}\right\}}^{\mathsf{\text{T}}}\right)$ as well as those of the numerical complexities and CPU time as a function of SNR of all the six methods are shown in Figure 10. The proposed ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA methods provide the best estimation results with moderate computational complexities and CPU time. Generally speaking, the ${\text{JD}}_{\text{LU}}^{+}$ICA algorithm offers the best performance/complexity tradeoff in this BSS experimental context.
Conclusions
We have proposed two methods, called ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$, in order to achieve the CP decomposition of seminonnegative semisymmetric threeway arrays. The nonnegativity constraint is imposed on the two symmetric modes of threeway arrays by means of a square change of variable, giving rise to an unconstrained joint diagonalization by congruence problem. Therefore, the nonnegative loading matrix can be estimated by computing the joint diagonalizer. We consider the elementary LU and QR parameterizations of the Hadamard square root of the nonnegative joint diagonalizer, leading to two Jacobilike optimization procedures. In each Jacobilike iteration, the optimization is formulated into a minimization of a polynomial or rational function with respect to only one parameter. In addition, the numerical complexity for each algorithm has been analyzed.
The performance of the proposed ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$ algorithms is evaluated with simulated seminonnegative semisymmetric threeway arrays. Four classical nonorthogonal JDC methods without nonnegativity constraints including ACDC [23], FFDIAG [25], LUJ1D [26], and QRJ1D [26] and one nonnegative JDC method ${\text{ACDC}}_{\text{LU}}^{+}$[41] are tested as reference methods. The performance is assessed in terms of the matrix estimation accuracy, the numerical complexity, and the CPU time. The convergence property, the influence of SNR, the impact of dimension, the effect of coherence, and the influence of condition number are extensively studied by Monte Carlo experiments. The obtained results show that the proposed algorithms offer better estimation accuracy by means of exploiting the nonnegativity a priori. The ${\text{JD}}_{\text{LU}}^{+}$ algorithm provides the best performance/complexity compromise.
The proposed algorithms are suitable tools for solving the ICA problems where a nonnegative mixing matrix is considered, such as in MRS. In this case, the threeway array built by stacking the matrix slices of a HO cumulant array fulfills the seminonnegative semisymmetric structure. We proposed two ICA methods, namely ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA, based on CP decomposition of the fourthorder cumulant array using ${\text{JD}}_{\text{LU}}^{+}$ and ${\text{JD}}_{\text{QR}}^{+}$, respectively. The source separation ability of the proposed algorithms is verified through a BSS application carried out on simulated MRS data. The ${\text{JD}}_{\text{LU}}^{+}$ICA and ${\text{JD}}_{\text{QR}}^{+}$ICA are compared to one NMF method [56], one nonnegative ICA method [55], and two classical ICA methods, namely CoM _{2}[54] and SOBI [47]. The performance is comprehensively studied as a function of the number of observations and of SNR. The experimental results demonstrate the improvement of the proposed methods in terms of the source estimation accuracy and also show that exploiting the two a priori of the data, namely the nonnegativity of the mixing matrix and the statistical independence of the sources, allows us to achieve better estimation results. The ${\text{JD}}_{\text{LU}}^{+}$ICA algorithm provides the best performance/complexity tradeoff.
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Wang, L., Albera, L., Kachenoura, A. et al. Canonical polyadic decomposition of thirdorder seminonnegative semisymmetric tensors using LU and QR matrix factorizations. EURASIP J. Adv. Signal Process. 2014, 150 (2014). https://doi.org/10.1186/168761802014150
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Keywords
 Canonical polyadic decomposition
 Seminonnegative semisymmetric tensor
 Joint diagonalization by congruence
 Individual differences in scaling analysis
 Blind source separation
 Independent component analysis
 Magnetic resonance spectroscopy